I'll first summarize my conjecture about the p-adic slopes of modular forms for GL_2 (both classical and overconvergent). This conjecture is based upon some structures in the geometry of the special fibers of elliptic modular curves at p. In an attempt to generalize to GL_n, I'll discuss the analogue of these structures (after accounting for p-torsion) in spaces of modular symbols for GL_2. And then finally I'll mention similar structures in the cohomology of symmetric power representations of SL_n(Z), and review methods for computing Hecke spectra in this setting (at least for the top cohomology groups)using the topology of the Tits building.